Ground State Solutions for Asymptotically Periodic Kirchhoff-Type Equations with Asymptotically Cubic or Super-cubic Nonlinearities

This paper is concerned with the following Kirchhoff-type equation

$$begin{aligned} -left( a+bint _{mathbb {R}^3}|nabla {u}|^2mathrm {d}xright) triangle u+V(x)u=f(x, u), quad xin mathbb {R}^{3}, end{aligned}$$

where (Vin mathcal {C}(mathbb {R}^{3}, (0,infty ))), (fin mathcal {C}({mathbb {R}}^{3}times mathbb {R}, mathbb {R})), V(x) and f(x, t) are periodic or asymptotically periodic in x. Using weaker assumptions (lim _{|t|rightarrow infty }frac{int _0^tf(x, s)mathrm {d}s}{|t|^3}=infty ) uniformly in (xin mathbb {R}^3) and

$$begin{aligned}&left[ frac{f(x,tau )}{tau ^3}-frac{f(x,ttau )}{(ttau )^3} right] mathrm {sign}(1-t) +theta _0V(x)frac{|1-t^2|}{(ttau )^2}ge 0, quad &quad forall xin mathbb {R}^3, t>0, tau ne 0 end{aligned}$$

with a constant (theta _0in (0,1)), instead of the common assumption (lim _{|t|rightarrow infty }frac{int _0^tf(x, s)mathrm {d}s}{|t|^4}=infty ) uniformly in (xin mathbb {R}^3) and the usual Nehari-type monotonic condition on (f(x,t)/|t|^3), we establish the existence of Nehari-type ground state solutions of the above problem, which generalizes and improves the recent results of Qin et al. (Comput Math Appl 71:1524–1536, 2016) and Zhang and Zhang (J Math Anal Appl 423:1671–1692, 2015). In particular, our results unify asymptotically cubic and super-cubic nonlinearities.